Mathematical modeling of separation

The Supplement B (reference) contains a description of the process to render an automatic construction of mathematical models with the application of electronic computer. The research work of the Institute of the applied mathematics of The Academy of Sciences ( Ukraine) was assumed as a basis for the Supplement. The prepared mathematical model provides the possibility to spare strength and to save money, usually spent for the development of the mathematical models of each separate enterprise. The model provides the possibility to execute the works standard forms and records for the non-destructive inspection in complete correspondence with the requirements of the Standard. 

Distillation Columns. Distillation is by far the most common separation technique in the chemical process industries. Tray and packed columns are employed as strippers, absorbers, and their combinations in a wide range of diverse appHcations. Although the components to be separated and distillation equipment may be different, the mathematical model of the material and energy balances and of the vapor—Hquid equiUbria are similar and equally appHcable to all distillation operations. Computation of multicomponent systems are extremely complex. Computers, right from their eadiest avadabihties, have been used for making plate-to-plate calculations. 

For field-oriented controls, a mathematical model of the machine is developed in terms of rotating field to represent its operating parameters such as /V 4, 7, and 0 and all parameters that can inlluence the performance of the machine. The actual operating quantities arc then computed in terms of rotating field and corrected to the required level through open- or closed-loop control schemes to achieve very precise speed control. To make the model similar to that lor a d.c. machine, equation (6.2) is further resolved into two components, one direct axis and the other quadrature axis, as di.sciis.sed later. Now it is possible to monitor and vary these components individually, as with a d.c. machine. With this phasor control we can now achieve a high dynamic performance and accuracy of speed control in an a.c. machine, similar to a separately excited d.c. machine. A d.c. machine provides extremely accurate speed control due to the independent controls of its field and armature currents. 

Obviously, construction of a mathematical model of this process, with our present limited knowledge about some of the critical details of the process, requires good insight and many qualitative judgments to pose a solvable mathematical problem with some claim to realism. For example what dictates the point of phase separation does equilibrium or rate of diffusion govern the monomer partitioning between phase if it is the former what are the partition coefficients for each monomer which polymeric species go to each phase and so on. 

What is commonly understood by a fundamental approach is applying theoretically based mathematical models of necessary equipment items. Intrinsic (not falsified by processes other than a chemical transformation) kinetics of all processes are investigated, transport phenomena are studied, flow patterns are identified, and relevant microscopic phenomena are studied. It is intended to separately study as many intrinsic stages as possible and to combine results of these investigations into a mathematical model. Such a model contains only a limited amount of theory (grey models, gross models, or tendency models). Obviously, the extrapolation power of these models strongly depends on the content of theory. The model... 

Molecular mechanics force fields rest on four fundamental principles. The first principle is derived from the Bom-Oppenheimer approximation. Electrons have much lower mass than nuclei and move at much greater velocity. The velocity is sufficiently different that the nuclei can be considered stationary on a relative scale. In effect, the electronic and nuclear motions are uncoupled, and they can be treated separately. Unlike quantum mechanics, which is involved in determining the probability of electron distribution, molecular mechanics focuses instead on the location of the nuclei. Based on both theory and experiment, a set of equations are used to account for the electronic-nuclear attraction, nuclear-nuclear repulsion, and covalent bonding. Electrons are not directly taken into account, but they are considered indirectly or implicitly through the use of potential energy equations. This approach creates a mathematical model of molecular structures which is intuitively clear and readily available for fast computations. The set of equations and constants is defined as the force... 

He has published over 200 papers in the fields of process control, optimization, and mathematical modeling of processes such as separations, combustion, and microelectronics processing. He is coauthor of Process Dynamics and Control, published by Wiley in 1989. Dr. Edgar was chairman of the CAST Division of AIChE in 1986, president of the CACHE Corporation from 1981 to 1984, and president of AIChE in 1997. 

One of the least well understood aspects of the whole field is the precise physical nature of the process whereby polymer chains of a different size are separated by passage through a gel column. On a qualitative level adec[uate explanations of the phenomenon exist but it has proved to be a more difficult task to formulate and solve anything other thcui the simplest of mathematical models of the chromatographic process. 

Pure pressure flow was first formulated and solved by Joseph Boussinesq in 1868, and combined pressure and drag flow in 1922 by Rowell and Finlayson (19) in the first mathematical model of screw-type viscous pumps. The detailed solution by the method of separation of variables is given elsewhere (17c), and the resulting velocity profile is given by... 

R. M. Counce, et ah, Mathematical model of sulfur dioxide absorption into a calcium hydroxide slurry in a spray dryer, Separation Sci. Technol.,... 

Mathematical models of biological processes are often used for hypothesis testing and process optimization. Using physical interpretation of results to obtain greater insight into process behavior is only possible when structured models that consider several parts of the system separately are employed. A number of dynamic mathematical models for cell growth and metabolite pro-... 

Such discrepancy is observed in many other cases [12, 13] as well as in cases of tumors growth considered above. Obviously, mathematical model of growth (8) is a very rough approximation. It may be used for rough estimation, for example for classification of population development [16, 17] and also for description of experimental data on separate sections of growth curves. 

It is important to notice that, from both viewpoints, as well as in all working procedures, experimental data are required and that, at the same time, mathematical models are absolutely needed for data processing. Generally, when the mathematical model of a process is relatively complex, a good accuracy and an important volume of experimental data are simultaneously required. Therefore, in these cases the quality of the determination of parameters is the most important factor to ensure model relevance. The strategy adopted in these cases is very simple for all the parameters of the process that accept an indirect identification, the research procedure of identification is carried out separately from the real process whereas for the very specific process parameters that are difficult to identify indirectly, experiments are carried out with the actual process. 

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